We report on the first dispersive calculation of the $\eta$ and $\eta'$ pole contributions [1,2] to hadronic light-by-light scattering in the anomalous magnetic moment of the muon, which, next to the pion pole [3,4], give rise to the leading singularities. The poles' strengths are unambiguously determined by the respective transition form factors. Several low- and high-energy physics constraints are utilized in our calculation. Our final data-driven results with fully controlled uncertainty estimates, $a_\mu^{\eta\text{-pole}}=14.7(9)\times 10^{-11}$ and $a_\mu^{\eta'\text{-pole}}=13.5(7)\times 10^{-11}$, conclude a dedicated effort to evaluate the pseudoscalar-pole contributions using dispersion relations, amounting to a combined $a_\mu^{\text{PS-poles}}=91.2^{+2.9}_{-2.4}\times 10^{-11}$.